In this subsection we recall some of the basic properties of Locally finitely presented categories, for details and more information the reader is referred to [1]}.
Recall that a non-empty
category \(\mathcal {I}\)  is said to be filtered provided that for each pair of objects \(\lambda _1,\lambda _2\in \mathcal {I}\)  there are morphisms \(\varphi _i:\lambda _i\rightarrow \mu \)  for some \(\mu \in \mathcal {I}\) , and for each pair of morphisms \(\varphi _1,\varphi _2:\lambda \rightarrow \mu \)  there is a morphism \(\psi :\mu \rightarrow \nu \)  with \(\psi \varphi _1=\psi \varphi _2\) . Let \(\mathcal {A}\)  be an additive category, \(\mathcal {I}\)  a small filtered category and \(X:\mathcal {I}\rightarrow \mathcal {A}\)  be an additive functor. As usual we use the term direct limit for colimit of \(X\)  when \(I\)  is filtered. And we denote it by \(\underrightarrow{\operatorname{Lim}}_{i\in \mathcal {I}}X_i\) .
